Embedded newform 74.5.g.a.23.5

• Label: 74.5.g.a.23.5
• Level: $74$
• Weight: $5$
• Character: 74.23
• Analytic conductor: $7.649$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 74 = 2 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	74.g (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.64937726820\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(7\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			23.5
Character	\(\chi\)	\(=\)	74.23
Dual form			74.5.g.a.29.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.732051 + 2.73205i) q^{2} +(4.95016 - 2.85797i) q^{3} +(-6.92820 + 4.00000i) q^{4} +(7.81136 - 29.1524i) q^{5} +(11.4319 + 11.4319i) q^{6} +(-41.9747 - 72.7023i) q^{7} +(-16.0000 - 16.0000i) q^{8} +(-24.1640 + 41.8532i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 28 q^{2} + 42 q^{5} - 40 q^{6} + 48 q^{7} - 448 q^{8} + 556 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/74\mathbb{Z}\right)^\times\).

\(n\)	\(39\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.732051	+	2.73205i	0.183013	+	0.683013i
							\(3\)	4.95016	−	2.85797i	0.550017	−	0.317553i	−0.199112	−	0.979977i	\(-0.563806\pi\)
0.749129	+	0.662424i	\(0.230472\pi\)
\(5\)	7.81136	−	29.1524i	0.312454	−	1.16610i	−0.613882	−	0.789398i	\(-0.710393\pi\)
							0.926336	−	0.376697i	\(-0.122940\pi\)
\(7\)	−41.9747	−	72.7023i	−0.856627	−	1.48372i	−0.875128	−	0.483892i	\(-0.839223\pi\)
							0.0185011	−	0.999829i	\(-0.494111\pi\)
\(11\)		−	212.290i		−	1.75446i	−0.480071	−	0.877230i	\(-0.659389\pi\)
							0.480071	−	0.877230i	\(-0.340611\pi\)
\(13\)	−58.0525	+	216.655i	−0.343506	+	1.28198i	0.550842	+	0.834609i	\(0.314307\pi\)
							−0.894348	+	0.447372i	\(0.852360\pi\)
\(17\)	318.118	−	85.2395i	1.10076	−	0.294946i	0.337683	−	0.941260i	\(-0.390357\pi\)
							0.763072	+	0.646313i	\(0.223690\pi\)
\(19\)	19.3741	−	72.3052i	0.0536679	−	0.200291i	−0.933886	−	0.357570i	\(-0.883605\pi\)
							0.987554	+	0.157279i	\(0.0502720\pi\)
\(23\)	312.857	+	312.857i	0.591413	+	0.591413i	0.938013	−	0.346600i	\(-0.112664\pi\)
							−0.346600	+	0.938013i	\(0.612664\pi\)
\(29\)	611.601	−	611.601i	0.727230	−	0.727230i	−0.242837	−	0.970067i	\(-0.578078\pi\)
							0.970067	+	0.242837i	\(0.0780780\pi\)
\(31\)	787.326	−	787.326i	0.819278	−	0.819278i	−0.166725	−	0.986003i	\(-0.553319\pi\)
							0.986003	+	0.166725i	\(0.0533193\pi\)
\(37\)	−1335.92	+	299.148i	−0.975833	+	0.218516i
							\(41\)	−1806.50	+	1042.98i	−1.07466	+	0.620455i	−0.929451	−	0.368946i	\(-0.879719\pi\)
−0.145209	+	0.989401i	\(0.546385\pi\)
\(43\)	1003.46	+	1003.46i	0.542704	+	0.542704i	0.924321	−	0.381617i	\(-0.124633\pi\)
							−0.381617	+	0.924321i	\(0.624633\pi\)
\(47\)	−2110.69			−0.955497			−0.477749	−	0.878497i	\(-0.658547\pi\)
							−0.477749	+	0.878497i	\(0.658547\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	74.5.g.a.23.5	✓	28
37.29	odd	12	inner	74.5.g.a.29.5	yes	28

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
74.5.g.a.23.5	✓	28	1.1	even	1	trivial
74.5.g.a.29.5	yes	28	37.29	odd	12	inner